• How do you distinguish between no information and uncertain information?

• How do you express your uncertainty in your initial guess?

• Express prior belief as a distribution rather than a single value.

• Call it your prior for short

• Prior distributions are your entire belief state
• Write it as \(\theta = \text{P(guilty)}\) (not just whether it broke it or not)

Beyesian statistics (generally) start with analyst-informed priors

Frequentist statistics:

• Hypothesis testing \(H_0\), \(H_a\), and p-values
• Ignore priors / always start with no prior information, i.e. the “flat prior”
• Interested in likelihood of a parameter given the data: \(P(\text{data} | \theta) = \mathcal L(\theta | \text{data})\)
• Not interested in finding the most likely value of \(\theta\)
• \(\mathcal L(\theta | \text{data})\) is a function, not a probability distribution

• Inspect the frame and the area around it:
• Summarize all new information in a new probability distribution: \(P(D|\theta)\)
• The likelihood:
  • Contains all information you gathered on your data, \(D\)
  • Gives \(P(D|\theta = \theta_i)\) for all possible values of \(\theta\)
• \(p\)-values are the probability of observing \(D\) or a more extreme one

• The nail gave out due to excessive weight
• Other things are knocked over

• Maybe the frame just fell and scared the cat?
• More probable if the cat is generally innocent.

• Ultimately, we want the posterior probability of the cat being guilty, \(P(\theta | D)\)
• Combine prior and likelihood using Bayes’ theorem:

\[\text{Posterior} = \frac{\text{Prior} \times \text{Likelihood}}{\text{constant}}\]

• Ultimately, we want the posterior probability of the cat being guilty, \(P(\theta | D)\)
• Combine prior and likelihood using Bayes’ theorem:

\[\text{Posterior} = \frac{\text{Prior} \times \text{Likelihood}}{\text{constant}}\]

• Can find the constant through some simulations (e.g., the Metropolis–Hastings algorithm)

CC BY 4.0: Maxwell B. Joseph

\[\text{Posterior} = \frac{\text{Prior} \times \text{Likelihood}}{\text{constant}}\]

\[\text{Posterior} = \frac{\text{Prior} \times \text{Likelihood}}{\text{constant}}\]

• Prior, likelihood, and posterior are probability distributions
• We can use them to express our uncertainty using intuitive measures of probability

Prior predictive intervals: range of credible (i.e., believable) values with some degree of credibility

E.g., 50% prior predictive intervals using 25% and 75% percentiles:

Prior predictive intervals: range of credible (i.e., believable) values with some degree of credibility

E.g., 50% prior predictive intervals using 10% and 60% percentiles:

Compatibility intervals: compatibility of the data with specific values of \(\theta\)

NOTE: this interval indicates a probability of the data, not of \(\theta\)!

E.g., 50% likelihood confidence interval using 25% and 75% percentiles:

Posterior credible intervals: range of credible (i.e., believable) values of \(\theta\) with some degree of credibility

E.g., 50% posterior credible intervals using 25% and 75% percentiles:

\(\bullet\) posterior mean

\(\textbf{-}\) posterior median

\(\color{forestgreen}{\text{green lines}}\) 50%, 70%, 90%, and 99% credible intervals